Gradient Descent on Logistic Regression with Non-Separable Data and Large Step Sizes
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916466591268864 |
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| author | Meng, Si Yi Orvieto, Antonio Cao, Daniel Yiming De Sa, Christopher |
| author_facet | Meng, Si Yi Orvieto, Antonio Cao, Daniel Yiming De Sa, Christopher |
| contents | We study gradient descent (GD) dynamics on logistic regression problems with large, constant step sizes. For linearly-separable data, it is known that GD converges to the minimizer with arbitrarily large step sizes, a property which no longer holds when the problem is not separable. In fact, the behaviour can be much more complex -- a sequence of period-doubling bifurcations begins at the critical step size $2/λ$, where $λ$ is the largest eigenvalue of the Hessian at the solution. Using a smaller-than-critical step size guarantees convergence if initialized nearby the solution: but does this suffice globally? In one dimension, we show that a step size less than $1/λ$ suffices for global convergence. However, for all step sizes between $1/λ$ and the critical step size $2/λ$, one can construct a dataset such that GD converges to a stable cycle. In higher dimensions, this is actually possible even for step sizes less than $1/λ$. Our results show that although local convergence is guaranteed for all step sizes less than the critical step size, global convergence is not, and GD may instead converge to a cycle depending on the initialization. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_05033 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Gradient Descent on Logistic Regression with Non-Separable Data and Large Step Sizes Meng, Si Yi Orvieto, Antonio Cao, Daniel Yiming De Sa, Christopher Machine Learning Optimization and Control We study gradient descent (GD) dynamics on logistic regression problems with large, constant step sizes. For linearly-separable data, it is known that GD converges to the minimizer with arbitrarily large step sizes, a property which no longer holds when the problem is not separable. In fact, the behaviour can be much more complex -- a sequence of period-doubling bifurcations begins at the critical step size $2/λ$, where $λ$ is the largest eigenvalue of the Hessian at the solution. Using a smaller-than-critical step size guarantees convergence if initialized nearby the solution: but does this suffice globally? In one dimension, we show that a step size less than $1/λ$ suffices for global convergence. However, for all step sizes between $1/λ$ and the critical step size $2/λ$, one can construct a dataset such that GD converges to a stable cycle. In higher dimensions, this is actually possible even for step sizes less than $1/λ$. Our results show that although local convergence is guaranteed for all step sizes less than the critical step size, global convergence is not, and GD may instead converge to a cycle depending on the initialization. |
| title | Gradient Descent on Logistic Regression with Non-Separable Data and Large Step Sizes |
| topic | Machine Learning Optimization and Control |
| url | https://arxiv.org/abs/2406.05033 |